Single-molecule Forster resonance energy transfer (FRET) measurements on single biological molecules can be used to measure the structural and dynamical properties of subpopulations, as well as the kinetics of transitions between subpopulations. The most frequent single molecule FRET experiment has been to collect photons emitted by donor and acceptor fluorophores attached to the molecule freely diffusing through the illuminated volume of a confocal microscope. The photons are binned, and a histogram of the FRET efficiencies for each bin, defined as the fraction of the photons emitted from the acceptor, is constructed. The shape of the histogram depends on the conformational states of the molecule and their interconversion rates. In 1 we developed a simple analytic theory to describe FRET efficiency histograms constructed from a photon trajectory generated by a molecule with multiple conformational states. The histograms are approximated by a sum of Gaussians with the parameters explicitly determined by the FRET efficiencies of the states and the rates of the transitions between the states. The theory, which has been tested against exact histograms for two conformational states and simulated data for three and four conformational states, accurately describes how the peaks in the histograms collapse as the bin time or the transition rates increase. In 2 this approach and a complimentary maximum likelihood approach we developed earlier were used to extract folding and unfolding rate coefficients from single-molecule FRET data for proteins with kinetics too fast to measure waiting time distributions. Mechanical forces exerted on single molecules by laser optical tweezers or atomic force microscopes can induce structural transitions such as the unfolding of a protein or nucleic acid or the dissociation of a complex. In these experiments, the molecular system can be driven far from equilibrium, permitting the exploration of metastable states and rare molecular processes. . In 2001 we showed how to reconstruct molecular free energy surfaces by extending the Jarzynski equality from a relation for the system free energy that depends on an experimentally controllable parameter to a relation for a molecular free energy surface that depends on a fluctuating coordinate. In 3 we show that unperturbed free energy profiles as a function of molecular extension can be obtained rigorously from such experiments without using work-weighted position histograms we introduced previously.. An inverse Weierstrass transform is used to relate the system free energy directly to the underlying molecular free energy surface. An accurate approximation for the free energy surface is obtained by using the method of steepest descent to evaluate the inverse transform. The formalism is applied to simulated data obtained from a kinetic model of RNA folding, in which the dynamics consists of jumping between linker-dominated folded and unfolded free energy surfaces. Substrate binding sites in enzymes are often buried. In many cases they are accessible from the surface of the macromolecule by a narrow tunnel. In order to reach the active site, a ligand must diffuse into and then through the crevice leading to the site. A classic example is acetylcholinesterase . Transmembrane channels also can have binding sites that are produced by a pore-lining residue that can covalently bind a permeant ion or molecule. In 4 we consider diffusion-influenced binding to a buried binding site that is connected to the surface by a narrow tunnel. Under the single assumption of an equilibrium distribution of ligands over the tunnel cross section, we reduce the calculation of the time-dependent rate coefficient to the solution of a one-dimensional diffusion equation with appropriate boundary conditions. We obtain a simple analytical expression for the steady-state rate that depends on the potential of mean force in the tuunnel and the diffusion-controlled rate of binding to the tunnel entrance. Potential applications of our theory include substrate binding to a buried active site of an enzyme and permeant ion binding to an internal site in a transmembrane channel. When there is a separation of time scales, an effective description of the dynamics of the slow variables can be obtained by adiabatic elimination of fast ones. For example, for anisotropic Langevin dynamics in two dimensions, the conventional procedure leads to a Langevin equation for the slow coordinate that involves the potential of the mean force. The friction constant along this coordinate remains unchanged. In 5, we show that a more accurate, but still Markovian, description of the slow dynamics can be obtained by using position-dependent friction that is related to the time integral of the autocorrelation function of the difference between the actual force and the mean force by a Kirkwood-like formula. The result is generalized to many dimensions, where the slow or reaction coordinate is an arbitrary function of the Cartesian coordinates. When the fast variables are effectively one-dimensional, the additional friction along the slow coordinate can be expressed in closed form for an arbitrary potential. For a cylindrically symmetric channel of varying cross section with winding centerline, our analytical expression immediately yields the multidimensional version of the Zwanzig-Bradley formula for the position-dependent diffusion coefficient.